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Chapter 0: Problem 56

In Exercises \(55-60\), find the distance between \(a\) and \(b\). $$ a=-126, b=-75 $$

Short Answer

Expert verified
The distance between \(a\) and \(b\) is 51

Step by step solution

01

Identify Numbers

The problem provides two numbers. Here, number \(a\) is -126 and number \(b\) is -75. The task is to find the distance between \(a\) and \(b\). To determine the distance, the absolute difference between \(a\) and \(b\) must be calculated.
02

Calculate Difference

Subtract the smaller number from the larger one. In this case, \(b - a\) or \(-75 - (-126)\).
03

Calculate Absolute Value

Calculate the absolute value of the difference, which represents the distance between the two numbers. This is denoted by \|b - a\| or \|-75 - (-126)| .

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Absolute Value
The concept of absolute value is a fundamental part of math, especially when calculating distances. Absolute value is denoted by vertical bars around a number, like \(|x|\). It represents how far that number is from zero on a number line, without considering direction. For instance, the absolute value of both 5 and -5 is 5, because both are five units away from zero.

In a distance calculation between two real numbers, absolute value helps simplify the difference by ignoring negative signs. This means when you have a difference between two numbers, say \(b - a\), the absolute value \(|b - a|\) ensures the distance is always a non-negative number.
Subtraction
Subtraction is the operation used to find the difference between two numbers. In our example with numbers \(-126\) and \(-75\), subtraction helps us determine how far these numbers are from each other on a number line.

Here are steps of subtraction to find the distance:
  • Identify which number is larger. In this case, \(-75\) is larger than \(-126\).
  • Subtract the smaller number from the larger number. For \(-75\) and \(-126\), calculate \(-75 - (-126)\).
This operation is crucial because it helps set up the calculation for absolute value, giving us the distance without any sign information.
Real Numbers
Real numbers include all the numbers you can find on the number line, both rational and irrational. In our exercise, \(-126\) and \(-75\) are negative real numbers. Understanding real numbers is key as it provides context for how subtraction works between positive and negative numbers.

With real numbers:
  • You can add, subtract, multiply, and divide them (except division by zero).
  • They always have a place on the number line, which helps in visualizing calculations like distance.
This visualization plays a major role when determining distances, as it gives a tangible way to see how numbers relate to each other spatially.

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Most popular questions from this chapter

Use a graphing utility to graph each function. Write a paragraph describing any similarities and differences you observe among the graphs. (a) \(y=x\) (b) \(y=x^{2}\) (c) \(y=x^{3}\) (d) \(y=x^{4}\) (e) \(y=x^{5}\) (f) \(y=x^{6}\)

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